On extremal service disciplines in single-stage queueing systems

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

Consideration is given to a G/G/1 queuing system where the objective is to maximize the probability that a customer's system time does not exceed a given deadline. The deadline is defined to be the maximum amount of time the customer can spend in the system. The authors show that for deadlines that are i.i.d. random variables with concave cumulative distribution functions, last-in-first-out (LIFO) gives the highest probability of success, and first-in-first-out (FIFO) the lowest, over the class of all work-conserving nonpreemptive service disciplines that are independent of service time and deadline. The results are extended to include policies that allow preemption in G/M/1 systems, deadlines imposed on queuing (rather than system) time, and multiple server queues. >

short-paper Share on Tail asymptotics of the stationary distribution for a two-node generalized Jackson network Author: Masakiyo Miyazawa Tokyo University of Science, Noda, Chiba, Japan Tokyo University of Science, Noda, Chiba, JapanView Profile Authors Info & Claims ACM SIGMETRICS Performance Evaluation ReviewVolume 42Issue 2September 2014pp 70–72https://doi.org/10.1145/2667522.2667545Published:04 September 2014Publication History 1citation26DownloadsMetricsTotal Citations1Total Downloads26Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Performance of static priority rules for shared facilities in a remanufacturing shop with disassembly and reassembly

We consider a queueing system with C customer classes under a nonpreemptive service discipline. The goal is to find gradient estimators for the stationary average sojourn time per customer of each class under admission control. Due to the service discipline, IPA (infinitesimal perturbation analysis) estimators are not applicable. We present the idea of harmonic gradient (HG) estimation, based on the Fourier decomposition of periodic functions. The canonical estimators can be used to obtain consistent estimators for all the control variables in a single run. However, the large number of values for each parameter required in the estimation can greatly affect the performance. We then describe the implementations of the phantom RPA (rare perturbation analysis) method. This method requires evaluating, in parallel, the dynamics of as many phantom systems as customers in each busy period. Since this number is random, the implementation of the method can be rather complex. We use the Fourier decomposition ideas to construct a hybrid estimator that we call the phantom HG method. We then give simulation results to compare the performance of the estimators and their complexity.

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E2(E3)/1, M/H2(H3)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.

In order to achieve the parallel transmission of multiple service types in flying ad hoc networks (FANET) with the limited wireless channel resource, and meet the requirement of high-reliability and ultra-low-delay for the highest priority service, a novel multi-priority queueing mechanism is proposed for channel threshold based multiple access (CTMA) protocol. Firstly, according to the principle of CTMA protocol, we model the mechanism as a multi-priority M/G/1 queueing system with server’s probabilistic vacations in terms of queueing theory. Then we propose two priority queueing schemes for different priority types in detail, and derive some performance metrics, such as packet’s average sojourn time, etc., for both schemes theoretically. Results are compared and verified through simulations finally. Results show that the preemptive-resume mechanism for the highest priority service in Scheme 2 gains a large improvement in the performance of the highest priority service at the expense of a little performance of other priority services.

A generalized SBBP/G/1 queue and its applications

Análisis del servicio de Urgencias aplicando teoría de líneas de espera

In the spirit of Little's law $$L=\lambda W$$L=źW and its extension $$H=\lambda G$$H=źG we use sample-path analysis to give a general conservation law. For queueing models the law relates the asymptotic average workload in the system to the conditional asymptotic average sojourn time and service times distribution function. This law generalizes previously obtained conservation laws for both single- and multi-server systems, and anticipating and non-anticipating scheduling disciplines. Applications to single- and multi-class queueing and other systems that illustrate the versatility of this law are given. In particular, we show that, for anticipative and non-anticipative scheduling rules, the unconditional delay in a queue is related to the covariance of service times and queueing delays.

Exact results are presented for the average customer waiting and sojourn times in systems comprising a number of finite-capacity queues cyclically attended by several servers. The system behavior is described using generalized stochastic Petri nets (GSPN), and the performance indices are obtained numerically by means of GreatSPN, a software tool for the analysis of GSPN. The average waiting and sojourn times of customers in Markovian multiserver multiunique systems are computed exactly for the first time; they were previously estimated by other authors using approximate models. >

Sensitivity analysis for output performance measures in long-range dependent queueing system

We analyzed an M/M/R queueing system with finite capacity N where customers have multiple classes with no priorities under steady-state conditions. It is assumed that any one class of arriving customer is serviced by one or more servers, and that each customer class has equal probability of service. Analytic explicit solutions are obtained with the assumption of exponential distributions for arrival times and service times. A cost model is developed to determine the optimal system capacity and the optimal number of servers. The minimum expected cost, the optimal system capacity, the optimal number of servers, and various system performance measures are obtained for three customer classes and four customer classes, based on assumed numerical values given to the designated system parameters.

In this paper a fluid approximation, also known as a functional strong law of large numbers (FSLLN) for a GI/G/1 queue under a processor-sharing service discipline is established and its properties are analysed. The fluid limit depends on the arrival rate, the service time distribution of the initial customers, and the service time distribution of the arriving customers. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the fluid limit is piecewise linear and depends on the service time distribution only through its mean. The piecewise linear form of the limit can be recovered by an equilibrium type choice of the initial service distribution.

BackgroundNational guidelines in most countries set screening intervals for diabetic retinopathy (DR) that are insufficiently informed by contemporary incidence rates. This has unspecified implications for interval disease risks (IDs) of referable DR, disparities in ID between groups or individuals, time spent in referable state before screening (sojourn time), and workload. We explored the effect of various screening schedules on these outcomes and developed an open-access interactive policy tool informed by contemporary DR incidence rates.Methods and findingsScottish Diabetic Retinopathy Screening Programme data from 1 January 2007 to 31 December 2016 were linked to diabetes registry data. This yielded 128,606 screening examinations in people with type 1 diabetes (T1D) and 1,384,360 examinations in people with type 2 diabetes (T2D). Among those with T1D, 47% of those without and 44% of those with referable DR were female, mean diabetes duration was 21 and 23 years, respectively, and mean age was 26 and 24 years, respectively. Among those with T2D, 44% of those without and 42% of those with referable DR were female, mean diabetes duration was 9 and 14 years, respectively, and mean age was 58 and 52 years, respectively. Individual probability of developing referable DR was estimated using a generalised linear model and was used to calculate the intervals needed to achieve various IDs across prior grade strata, or at the individual level, and the resultant workload and sojourn time. The current policy in Scotland—screening people with no or mild disease annually and moderate disease every 6 months—yielded large differences in ID by prior grade (13.2%, 3.6%, and 0.6% annually for moderate, mild, and no prior DR strata, respectively, in T1D) and diabetes type (2.4% in T1D and 0.6% in T2D overall). Maintaining these overall risks but equalising risk across prior grade strata would require extremely short intervals in those with moderate DR (1–2 months) and very long intervals in those with no prior DR (35–47 months), with little change in workload or average sojourn time. Changing to intervals of 12, 9, and 3 months in T1D and to 24, 9, and 3 months in T2D for no, mild, and moderate DR strata, respectively, would substantially reduce disparity in ID across strata and between diabetes types whilst reducing workload by 26% and increasing sojourn time by 2.3 months. Including clinical risk factor data gave a small but significant increment in prediction of referable DR beyond grade (increase in C-statistic of 0.013 in T1D and 0.016 in T2D, both p < 0.001). However, using this model to derive personalised intervals did not have substantial workload or sojourn time benefits over stratum-specific intervals. The main limitation is that the results are pertinent only to countries that share broadly similar rates of retinal disease and risk factor distributions to Scotland.ConclusionsChanging current policies could reduce disparities in ID and achieve substantial reductions in workload within the range of IDs likely to be deemed acceptable. Our tool should facilitate more rational policy setting for screening.